Comments on: Applied knot theory http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/ Because my LiveJournal is too silly Sat, 17 Sep 2011 02:28:35 +0000 hourly 1 http://wordpress.org/?v=3.2.1 By: John Armstrong http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160641 John Armstrong Mon, 16 Mar 2009 12:16:47 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160641 That's one way to look at it, yes. There are other ways to approach the "coloring matrix", though, which has given me no end of trouble in trying to publish this stuff. Not that it matters much anymore... That’s one way to look at it, yes. There are other ways to approach the “coloring matrix”, though, which has given me no end of trouble in trying to publish this stuff. Not that it matters much anymore…

]]> By: Michi http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160619 Michi Mon, 16 Mar 2009 05:37:14 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160619 John: You did. I just didn't go back and reread them when I sat down with this. I blame my head cold. So, you're basically saying that what I'm doing here is working in the decategorification of the categorification of the knot invariant? :-) John: You did. I just didn’t go back and reread them when I sat down with this. I blame my head cold.

So, you’re basically saying that what I’m doing here is working in the decategorification of the categorification of the knot invariant? :-)

]]> By: John Armstrong http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160618 John Armstrong Mon, 16 Mar 2009 05:23:59 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160618 Didn't I make a whole big post about colorings? Maybe even a series of them. The important thing is the set of colorings for a given tangle diagram. As you apply tangle moves (which generalize Reidemeister moves) you get <em>explicit bijections</em> between the sets of colorings, making the coloring set a knot <em>co</em>variant. But it's not just the set of colorings that matters, but what colorings these induce on the ends of the tangle. So what does this give us? A <em>span</em> of coloring sets! Then you can decategorify the spans to get a coloring matrix. What you showed is that two matrix entries don't agree, so the tangles can't be ambient-isotopic. Didn’t I make a whole big post about colorings? Maybe even a series of them.

The important thing is the set of colorings for a given tangle diagram. As you apply tangle moves (which generalize Reidemeister moves) you get explicit bijections between the sets of colorings, making the coloring set a knot covariant. But it’s not just the set of colorings that matters, but what colorings these induce on the ends of the tangle. So what does this give us? A span of coloring sets! Then you can decategorify the spans to get a coloring matrix. What you showed is that two matrix entries don’t agree, so the tangles can’t be ambient-isotopic.

]]> By: Michi http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160607 Michi Mon, 16 Mar 2009 03:07:20 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160607 If: that's unfortunate. Any idea what I could do to make it work? If: that’s unfortunate. Any idea what I could do to make it work?

]]> By: lf http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160606 lf Mon, 16 Mar 2009 03:03:42 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160606 Safari is cutting off the bottom half of each figure, making the type III moves rather confusing (unless you zoom out a few steps) :-( Safari is cutting off the bottom half of each figure, making the type III moves rather confusing (unless you zoom out a few steps) :-(

]]> By: Michi http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160604 Michi Mon, 16 Mar 2009 01:58:15 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160604 Thank you - I was a bit at a loss how to really describe the situation I was in. I think "framed links" was a bit of a cop out. Your lack of opposition to my proof warms me towards thinking it's actually correct, though, making this my first original result in Knot theory (original with the basis used for the Junior congresses: discovered independently by myself, regardless of whether it is an already known result or not). Thank you – I was a bit at a loss how to really describe the situation I was in. I think “framed links” was a bit of a cop out.

Your lack of opposition to my proof warms me towards thinking it’s actually correct, though, making this my first original result in Knot theory (original with the basis used for the Junior congresses: discovered independently by myself, regardless of whether it is an already known result or not).

]]> By: John Armstrong http://blog.mikael.johanssons.org/archive/2009/03/applied-knot-theory/comment-page-1/#comment-160603 John Armstrong Mon, 16 Mar 2009 01:52:39 +0000 http://blog.mikael.johanssons.org/?p=196#comment-160603 Of course, I have to jump in and point out that you're using colored <em>tangles</em>, not just knots. Of course, I have to jump in and point out that you’re using colored tangles, not just knots.

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